Method for determining live load vertical deformation of suspension bridge with horizontal cable and central buckle

Abstract

The application discloses a live load vertical deformation determination method for a suspension bridge with a horizontal cable and a central buckle, which comprises the following steps: 1, a central buckle is added; 2, a basic linear shape of a main cable under dead load considering bending stiffness is determined; 3, equivalent spring stiffness of side cables and the central buckle is calculated; 4, a main cable compatibility equation is established; 5, a deflection and beam end rotation angle analytical expression is established; 6, an unknown coefficient function is established; 7, a horizontal increment of the main cable on both sides of the central buckle caused by live load is solved; and 8, deflection and beam end rotation angle are solved. The application can greatly improve the vertical stiffness of the suspension bridge without changing the stress of the stiffening beam, so that the suspension bridge has stronger crossing capacity. The contribution of the central buckle to the overall vertical stiffness of the suspension bridge can be quickly evaluated, and the mechanical principle of the central buckle in the suspension bridge is described in an analytical way. The application is convenient to use, accurate and reliable in result, and can be used for suspension bridge structure analysis and preliminary design.

Description

Technical Field

[0001] This invention relates to the field of bridge analysis theory, specifically a method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle under live load. Background Technology

[0002] Suspension bridges are the preferred solution for crossing wide rivers or straits due to their superior spanning capacity and efficient material utilization, and the world's longest suspension bridge span is constantly being broken. Japan's Akashi Kaikyo Bridge was once the longest suspension bridge with a span of 1991m, until it was surpassed in 2022 by Turkey's Çanakkale Bridge (1915) (2023m). Many more magnificent bridges are under construction, including China's Shiziyang Bridge (2180m) and Zhangjinggao Yangtze River Bridge (2300m). Furthermore, the planned Messina Strait Bridge project in Italy has a span of 3300m.

[0003] However, whether the vertical stiffness of suspension bridges meets the deflection-to-span ratio design requirements has always been a hot research topic in engineering and academia. With increasing span, suspension bridges gradually struggle to meet the vertical stiffness requirements to resist the deformation of the stiffening girder caused by live loads. Therefore, the main span of current large-span suspension bridges is much smaller than the theoretically limit span corresponding to their material properties. The main reason for insufficient vertical stiffness in suspension bridges is not only the low bending stiffness of the stiffening girder, but also the large longitudinal displacement of the main cable under half-load, which increases the vertical displacement of the stiffening girder. This directly hinders the development of suspension bridges towards larger spans. The most direct way to improve the overall vertical stiffness of suspension bridges is to increase the area of ​​the main cable and the bending stiffness of the stiffening girder, but this will directly reduce the economic efficiency of the suspension bridge. Many new structural measures have been proposed to increase the vertical stiffness of suspension bridges, including double-main-cable designs with different cross-sections to reduce the displacement of asymmetrical deformation of the main cable, and using horizontal cables at the top of the bridge towers to constrain the deformation of the towers, which can also reduce the vertical deflection of the stiffening girder.

[0004] However, the aforementioned methods not only increase construction difficulty and affect the aesthetic appearance of the suspension bridge, but also significantly increase construction costs. Furthermore, while the central tie can reduce the relative displacement between the main cable and the stiffening girder, thus improving cable fatigue and wind resistance, the maximum deflection of the stiffening girder can be reduced through tower-girder consolidation and the use of a rigid central tie. This can lead to an imbalance in the tensile forces on either side of the central tie, affecting the structural safety of the stiffening girder. Therefore, the insufficient overall vertical stiffness of suspension bridges has consistently hindered the development of suspension bridges with larger spans. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle under live load. This method can quickly and accurately assess the contribution of the improved central buckle to the vertical stiffness of the suspension bridge, and is a method for calculating the structural deformation of the suspension bridge under live load with clear physical meaning.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle under live load includes the following steps.

[0008] Step 1: Adding a central buckle: A suspension bridge has at least one main cable, and each main cable is connected to a stiffening girder by several parallel suspenders; an inverted V-shaped central buckle is set at the mid-span of each main cable; the apex of the central buckle is anchored to the mid-span of the corresponding main cable, and the two bottom ends of the central buckle are anchored to the stiffening girder directly below; the stiffening girders on both sides of each central buckle are tensioned and constrained by horizontal cables.

[0009] Step 2: Determine the basic alignment of the main cable under dead load considering bending stiffness: Considering the bending stiffness of the main cable, and based on the main cable equilibrium equation under dead load, obtain the basic alignment y of the main cable under dead load.

[0010] Step 3: Calculate the equivalent spring stiffness of the side cable and the center buckle: The equivalent spring stiffness of the side cable is obtained based on the conservation of stress-free length; the equivalent spring stiffness K of the center buckle is... g It is calculated based on the longitudinal displacement of the midpoint of the main cable, and K g For H q1 H q2 The function of the stiffening beam deflection w; where H q1 and H q2 These are the horizontal increments of the main cable on both sides of the central buckle caused by live load, which are the quantities to be solved.

[0011] Step 4: Establish the main cable compatibility equations: Based on the deformation relationship of the main cables on both sides of the central buckle in the horizontal projection direction, establish the compatibility equations at both ends of the main cable; where both compatibility equations are H... q1 H q2 and K g The functional equation;

[0012] Step 5: Establish analytical expressions for deflection and beam end rotation: The starting point of the live load applied to the left side of the central buckle is x1, and the length of the live load section on the left side is t1; the starting point of the live load applied to the right side of the central buckle is x2, and the length of the live load section on the right side is t2; according to the live load position on the stiffening girder, the main span is divided into 6 continuous segments, namely: 0≤x≤x1, x1≤x≤x1+t1, x1+t1≤x≤L / 2, L / 2≤x≤L / 2+x, L / 2+x2≤x≤L / 2+x2+t2 and L / 2+x2+t2≤x≤L; where x is any point on the stiffening girder along the longitudinal direction of the bridge within the entire span; L is the total span length; due to the asymmetry of the horizontal forces of the main cables on both sides of the central buckle, the stiffening girder has a vertical equilibrium equation for each of the 6 segments of the main span; based on the 6 vertical equilibrium equations, 6 analytical expressions for the stiffening girder deflection are obtained; each analytical expression for the stiffening girder deflection is H q1 and H q2 The function has 4 unknown coefficients; the analytical expression for the beam end rotation angle is obtained by differentiating the analytical expression for deflection.

[0013] Step 6: Establish the unknown coefficient functions: Based on the characteristics that the vertical displacement at the stiffening beam end is 0, the bending moment at the stiffening beam end is 0, the displacement and rotation of the stiffening beam remain continuous, and the shear force and bending moment of the stiffening beam remain continuous, 24 boundary conditions are obtained. Then, the 24 unknown coefficient functions of the 6 deflection analytical expressions in Step 5 are solved; each unknown coefficient function is H. q1 and / or H q2 The function.

[0014] Step 7: Solve for H q1 and H q2 Specifically, it includes the following steps:

[0015] Step 7-1, Given H q1 and H q2 The initial value.

[0016] Step 7-2, Solve for the unknown coefficients: The H coefficients from step 7-1... q1 and H q2 The values ​​are substituted into the unknown coefficient function in step 6 to obtain 24 unknown coefficients.

[0017] Step 7-3, Solve for the stiffened beam deflection w: The H value from step 7-1... q1 and H q2 The value, along with the 24 unknown coefficients obtained from the steps, are substituted into the analytical expression for the stiffening beam deflection in step 5 to obtain the stiffening beam deflection w.

[0018] Step 7-4: Update the main cable compatibility equation: Change K in step 3... gSubstituting the expression and the stiffening girder deflection w obtained in step 7-3 into the main cable compatibility equation in step 4, we obtain the updated main cable compatibility equation; at this point, the updated main cable compatibility equation contains only the quantity to be solved, H. q1 H q2 The equation.

[0019] Step 7-5, Solve for H q1 and H q2 The manta ray foraging optimization algorithm is used to solve the updated main cable compatibility equation. When the updated main cable compatibility equation converges, H is obtained. q1 and H q2 The optimal value.

[0020] Step 8: Solve for deflection and beam end rotation: Based on H obtained in Step 7 q1 and H q2 Find the optimal value, repeat steps 7-2 and 7-3 to obtain the stiffening beam deflection, and differentiate the stiffening beam deflection to obtain the beam end rotation angle.

[0021] The apex of the central buckle and the corresponding trough of the main cable form anchorage point A. The two bottom endpoints of the central buckle and the stiffening beam directly below form left anchorage point B and right anchorage point C. The two horizontal cables are the left horizontal cable and the right horizontal cable. The left end of the left horizontal cable is anchored to the left bridge tower crossbeam, forming anchorage point D. The right end of the left horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point B. The left end of the right horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point C. The right end of the right horizontal cable is anchored to the right bridge tower crossbeam, forming anchorage point E.

[0022] In step 2, the formula for calculating the basic alignment y of the main cable under constant load is:

[0023]

[0024] in:

[0025]

[0026] In the formula, g g and g c These are the uniformly distributed self-weight loads of the stiffening girder and the main cable, respectively, both of which are known quantities.

[0027] H g The horizontal tension of the main cable under constant load is an unknown quantity, which is determined by boundary conditions.

[0028] K1, K2, K3, and K4 are all coefficients to be determined, which are obtained through boundary conditions;

[0029] E c and I c These are the elastic modulus and moment of inertia of the main cable, respectively.

[0030] 'a' represents intermediate computational quantities.

[0031] In step 2, solve for K1, K2, K3, K4 and H. g The five boundary conditions are as follows:

[0032]

[0033] In the formula, h is the mid-span elevation; x = 0 represents the hinge point between the left side of the main cable and the left bridge tower; x = L / 2 represents the mid-span point, and L / 2 is the half-span length; x = L represents the hinge point between the right side of the main cable and the right bridge tower.

[0034] In step 3, the side cables include a left cable located to the left of the main cable and a right cable located to the right of the main cable; the bending stiffness of the left cable is K. cl The bending stiffness of the right cable is K. cr The equivalent spring stiffness of the central buckle is K. g The corresponding formulas are as follows:

[0035]

[0036]

[0037]

[0038] in:

[0039]

[0040]

[0041] Δ=(w(L / 2-l h / 2)-w(L / 2+l h / 2)) / 2

[0042] In the formula, α is the inclination angle of the left or right cable.

[0043] L sl It is the horizontal projection length of the left cable; L sr It is the horizontal projection length of the cable on the right.

[0044] H q1 and H q2 These are the horizontal increments of the main cable on both sides of the central buckle caused by live load, which are the quantities to be solved.

[0045] A c It is the cross-sectional area of ​​the main cable.

[0046] l v and l hIt is the projected length of the center in both the vertical and horizontal directions.

[0047] E cb and A cb These are the elastic modulus and cross-sectional area of ​​the central buckle, respectively.

[0048] E hc and A hc These are the elastic modulus and cross-sectional area of ​​the horizontal cable, respectively.

[0049] δ c1 It is the longitudinal displacement of the central buckle under the action of the horizontal force of the main cable.

[0050] δ c2 The longitudinal displacement of the main cable mid-span is caused by the asymmetrical vertical displacement of the stiffening girder on both sides of the central buckle.

[0051] Δ represents the vertical displacement of the two sides of the central buckle.

[0052] w is the deflection of the stiffened beam.

[0053] In step 4, the compatibility equations for the main cables at both ends of the main span are:

[0054]

[0055]

[0056]

[0057]

[0058] In the formula, L c1 and L c2 These are all intermediate calculations.

[0059] w is the deflection of the stiffened beam, a quantity to be solved.

[0060] K tl and K tr These are the equivalent spring stiffnesses of the left and right bridge towers, respectively, and are known quantities.

[0061] In step 5, the stiffening beam deflection w includes six components: w1, w2, w3, w4, w5, and w6. Their specific analytical expressions are as follows:

[0062]

[0063] in:

[0064]

[0065]

[0066]

[0067]

[0068] κ1=E g I g +E c I c

[0069] κ2=(H g +H q1 (K3+K4) / 2-E c I c a(K3+K4) / 2

[0070] κ3=(H g +H q1 (-K3+K4) / 2-E c I c a(-K3+K4) / 2

[0071] κ4=(H g +H q2 (K3+K4) / 2-E c I c a(K3+K4) / 2

[0072] κ5=(H g +H q2 (-K3+K4) / 2-E c I c a(-K3+K4) / 2

[0073] In the formula, C1-C 24 There are 24 unknown coefficients; η1-η 12 There are 12 intermediate computational resources; κ1-κ5 are 5 intermediate computational resources; E g and I g These are the elastic modulus and moment of inertia of the stiffened beam, respectively; q is the uniformly distributed live load value.

[0074] In step 6, the 24 boundary conditions are as follows:

[0075] A. When the vertical displacement at the end of the stiffened beam is 0, the following two boundary conditions apply:

[0076] w1(0)=w6(L)=0

[0077] B. When the bending moment at the end of the stiffened beam is 0, the following two boundary conditions apply:

[0078] E g I g w″1(0)=E g I gw″6(L)=0

[0079] C. When the stiffened beam's displacement and rotation remain continuous, the following ten boundary conditions apply:

[0080] w1(x1)=w2(x1)

[0081] w′1(x1)=w′2(x1)

[0082] w2(x1+t1)=w3(x1+t1)

[0083] w′2(x1+t1)=w′3(x1+t1)

[0084] w3(L / 2)=w4(L / 2)

[0085] w′3(L / 2)=w′4(L / 2)

[0086] w4(L / 2+x2)=w5(L / 2+x2)

[0087] w′4(L / 2+x2)=w′5(L / 2+x2)

[0088] w5(L / 2+x2+t2)=w6(L / 2+x2+t2)

[0089] w′5(L / 2+x2+t2)=w′6(L / 2+x2+t2)

[0090] D. When the shear force and bending moment of the stiffened beam remain continuous, the following ten boundary conditions apply:

[0091] w″1(x1)=w″2(x1)

[0092] w″′1(x1)=w″′2(x1)

[0093] w″2(x1+t1)=w″3(x1+t1)

[0094] w″′2(x1+t1)=w″′3(x1+t1)

[0095] w″3(L / 2)=w″4(L / 2)

[0096] w″′3(L / 2)=w″′4(L / 2)

[0097] w″4(L / 2+x2)=w″5(L / 2+x2)

[0098] w″′4(L / 2+x2)=w″′5(L / 2+x2)

[0099] w″5(L / 2+x2+t2)=w″6(L / 2+x2+t2)

[0100] w″′5(L / 2+x2+t2)=w″′6(L / 2+x2+t2).

[0101] In step 7-1, H q1 and H q2 The method for giving the initial value is: H q1 =H q2 =qL 2 (t1+t2) / h; where h is the sag height at the mid-span.

[0102] The present invention has the following beneficial effects: The present invention proposes to improve the overall vertical stiffness of suspension bridges by improving the central buckle measures, which not only improves the stress of the stiffening girder, but also has extremely high economic efficiency; The proposed analytical calculation method clarifies the working principle of the improved central buckle, and the physical meaning is also clearer, which makes it easier to evaluate the contribution of the improved central buckle to the overall vertical stiffness of the suspension bridge, and can be applied to the preliminary design of suspension bridges. Attached Figure Description

[0103] Figure 1 This is a schematic diagram of the deformation of a suspension bridge under live load in a specific embodiment.

[0104] Figure 2 This is a schematic diagram illustrating the deformation of the improved central suspension bridge under live load conditions in a specific embodiment.

[0105] Figure 3 This is a schematic diagram of the improved central buckle elevation in a specific embodiment.

[0106] Figure 4 This is a three-dimensional schematic diagram of the improved central buckle in a specific embodiment.

[0107] Figure 5 This is a schematic diagram of the deformation of the side cable under live load in a specific embodiment.

[0108] Figure 6 This is a schematic diagram of the deformation of the improved central buckle under live load in a specific embodiment. Figure 1 .

[0109] Figure 7 This is a schematic diagram of the deformation of the improved central buckle under live load in a specific embodiment. Figure 2 .

[0110] Figure 8 This is a simplified schematic diagram of the main cable in a specific embodiment.

[0111] Figure 9 This is a schematic diagram of the force transmission of the main cable, suspenders, and stiffening beam in a specific embodiment.

[0112] Figure 10This is a schematic diagram of an equivalent simply supported beam of a suspension bridge under live load in a specific embodiment.

[0113] Among them are:

[0114] 10. Main cable; 20. Stiffening girder; 30. Central buckle; 40. Horizontal cable;

[0115] 50. Left bridge tower; 51. Lower crossbeam of the left bridge tower;

[0116] 60. Right-side bridge tower; 61. Lower crossbeam of the right-side bridge tower;

[0117] 70. Hanging rod. Detailed Implementation

[0118] The present invention will now be described in further detail with reference to the accompanying drawings and specific preferred embodiments.

[0119] In the description of this invention, it should be understood that the terms "left side," "right side," "upper part," "lower part," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. "First," "second," etc., do not indicate the importance of the components, and therefore should not be construed as a limitation of this invention. The specific dimensions used in this embodiment are only for illustrating the technical solution and do not limit the scope of protection of this invention.

[0120] A method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle under live load includes the following steps.

[0121] Step 1: Add a central buckle

[0122] like Figure 1 and Figure 4 As shown, the suspension bridge has at least one main cable 10, and in this embodiment, it is preferred to have two main cables.

[0123] like Figures 2 to 4 As shown, each main cable is connected to the stiffening beam 20 by several parallel suspenders 70; an inverted V-shaped central buckle 30 is provided at the midpoint of each main cable span.

[0124] The apex of the central buckle is anchored to the mid-span of the corresponding main cable through cable clamps, and the two bottom ends of the central buckle are anchored to the stiffening beam directly below at adjacent hanger positions, thus forming a triangular truss with the stiffening beam segment directly below.

[0125] The stiffening girders on both sides of each central buckle are tensioned and constrained by pre-tensioned horizontal cables. The number of horizontal cables is determined based on the load and structural parameters.

[0126] like Figure 3 and Figure 4 As shown, the apex of the central buckle and the corresponding linear trough of the main cable form anchorage point A. The two bottom endpoints of the central buckle and the stiffening beam directly below form left anchorage point B and right anchorage point C. The two horizontal cables are the left horizontal cable and the right horizontal cable. The left end of the left horizontal cable is anchored to the left bridge tower crossbeam, forming anchorage point D. The right end of the left horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point B. The left end of the right horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point C. The right end of the right horizontal cable is anchored to the right bridge tower crossbeam, forming anchorage point E.

[0127] like Figure 1 As shown, under the action of live load half span loading, the deflection generated by the stiffening girder of the suspension bridge is close to the peak value; the large deformation of the main cable is manifested in the vertical deformation and the longitudinal deformation of the bridge; while most of the stiffening girders of long-span suspension bridges have released the longitudinal constraints at the beam ends to improve the stress of the stiffening girder, so the stiffening girder will generate longitudinal displacement in the same direction as the main cable under the action of the suspenders.

[0128] like Figure 2 As shown, after the improved central buckle was installed, the longitudinal displacement of the main cable at mid-span caused by asymmetrical load was constrained by the rigid central buckle and the horizontal cables. At the same time, the horizontal increment difference of the main cables on both sides of the central buckle was borne by the horizontal cables under the stiffening girder. Since the improved central buckle effectively constrained the longitudinal displacement of the main cable caused by half-span loading, the vertical displacement of the stiffening girder and the main cable was greatly reduced, thereby improving the overall vertical stiffness of the suspension bridge.

[0129] In this embodiment, the rigid central buckle is preferably an I-beam structure, the cross-sectional dimensions and length of which are determined by the geometric positions of the stiffening beam, main cable and sling, and the live load design value.

[0130] Furthermore, the cross-sectional type of the horizontal cables is preferably the same as that of the main cables, and the specific number, preload and cross-sectional dimensions are determined by the specific structural parameters and live load design values ​​of the cable bridge.

[0131] Step 2: Determine the basic alignment of the main cable under dead load, taking into account bending stiffness.

[0132] (1) Considering the bending stiffness of the main cable, the equilibrium equation of the main cable under dead load is...

[0133]

[0134] In the formula: E c I c It is the bending stiffness of the main cable; where E c and I c These are the elastic modulus and moment of inertia of the main cable, respectively.

[0135] H g The horizontal tension of the main cable under constant load is an unknown quantity, which is determined by boundary conditions.

[0136] g g and g c These are the uniformly distributed self-weight loads of the stiffening girder and the main cable, respectively, both of which are known quantities.

[0137] (2) Cable alignment under constant load:

[0138]

[0139] in:

[0140]

[0141] In the formula, 'a' represents an intermediate calculation quantity.

[0142] K1, K2, K3, and K4 are all coefficients to be determined, which are obtained through boundary conditions;

[0143] (3) The coordinates of the mid-span of the suspension bridge are determined based on the design elevation. Assuming that the main cable is hinged to the bridge towers on both sides, the coordinates of K1, K2, K3, K4, and H are then calculated. g The five boundary conditions are as follows:

[0144]

[0145] In the formula, h is the mid-span elevation; x = 0 represents the hinge point between the left side of the main cable and the left bridge tower; x = L / 2 represents the mid-span point, and L / 2 is the half-span length; x = L represents the hinge point between the right side of the main cable and the right bridge tower.

[0146] Step 3: Calculate the equivalent spring stiffness of the side cable and the central buckle.

[0147] (1) As Figure 5 As shown, the equivalent spring stiffness of the side cable is obtained based on the stress-free length conservation.

[0148] The side cables include the left-side cable located to the left of the main cable and the right-side cable located to the right of the main cable; the bending stiffness of the left-side cable is K. cl The bending stiffness of the right cable is K. cr The equivalent spring stiffness of the central buckle is K. g The corresponding formulas are as follows:

[0149]

[0150]

[0151] In the formula, α is the inclination angle of the left or right cable.

[0152] L slIt is the horizontal projection length of the left cable; L sr It is the horizontal projection length of the cable on the right.

[0153] A c It is the cross-sectional area of ​​the main cable.

[0154] (2) Figure 6 As shown, the equivalent spring stiffness K of the improved central buckle g Based on the longitudinal displacement calculation of the mid-span of the main cable, the longitudinal displacement δ of the improved central buckle under the action of the horizontal force of the main cable is calculated. c1 for:

[0155]

[0156] In the formula, H q1 and H q2 These are the horizontal increments of the main cable on both sides of the central buckle caused by live load, which are the quantities to be solved.

[0157] l v and l h It is the projected length of the center in both the vertical and horizontal directions.

[0158] E cb and A cb These are the elastic modulus and cross-sectional area of ​​the central buckle, respectively.

[0159] E hc and A hc These are the elastic modulus and cross-sectional area of ​​the horizontal cable, respectively.

[0160] like Figure 7 As shown, the asymmetrical vertical displacement on both sides of the stiffening girder's central buckle causes the longitudinal displacement δ at the mid-span of the main cable. c2 for:

[0161]

[0162] in:

[0163] Δ=(w(L / 2-l h / 2)-w(L / 2+l h / 2)) / 2

[0164] In the formula: Δ is the vertical displacement of the two sides of the central buckle.

[0165] w is the deflection of the stiffened beam.

[0166] Improved equivalent spring stiffness K of the central buckle g The value is calculated based on the longitudinal displacement of the midpoint of the main cable span, and the specific expression is as follows:

[0167]

[0168] Therefore, K g For H q1 H q2 The function of the stiffening beam deflection w.

[0169] Step 4: Establish the main cable compatibility equation

[0170] like Figure 8 As shown, the main cable compatibility equation is used to determine the horizontal force increment of the main cable by coordinating the projected length of the main cable elongation in the horizontal direction.

[0171] Based on the deformation relationship of the main cables on both sides of the central buckle in the horizontal projection direction, the compatibility equations at both ends of the main cables are established; where both compatibility equations are H q1 H q2 and K g The functional equation is as follows:

[0172]

[0173]

[0174]

[0175]

[0176] In the formula, L c1 and L c2 These are all intermediate calculations.

[0177] w is the deflection of the stiffened beam, a quantity to be solved.

[0178] K tl and K tr These are the equivalent spring stiffnesses of the left and right bridge towers, respectively, and are known quantities.

[0179] Step 5: Establish analytical expressions for deflection and beam end rotation.

[0180] like Figure 9 As shown, the equilibrium equation of the main cable under live load is first established, and the force transmission path between the main cable, suspenders and stiffening girder is analyzed. Finally, the vertical equilibrium equation of the stiffening girder is established.

[0181] (1) The equilibrium equation of the main cable is:

[0182] E c I c (w″″+y″″)-(H g +H q (w″+y″)-g c -g g -s=0

[0183] Where s is the tension of the suspender membrane.

[0184] Rearranging the above equation, we can obtain the expression for the tension of the suspender membrane:

[0185] s = E c I c (w″″+y″″)-(H g +H q (w″+y″)-g c -g g

[0186] (2) The equilibrium equations for the stiffened beam are:

[0187] E g I g w″″=qs

[0188] Among them, E g and I g These are the elastic modulus and moment of inertia of the stiffened beam, respectively; q is the uniformly distributed live load value.

[0189] Substituting the tension of the suspender membrane into the above equation yields the vertical equilibrium equation for the stiffening beam:

[0190] E g I g w″″+E c I c (w″″+y″″)-(H g +H q (w″+y″)-g c -g g -q=0

[0191] Since the horizontal forces on the main cables on both sides of the improved central buckle are different under asymmetrical live loads, and in order to improve the versatility of the deformation determination method for suspension bridges with improved central buckles, the deformation of the stiffening girder under more load conditions is analyzed. Therefore, the load arrangement is as follows: live loads are applied to both sides of the central buckle. The starting point for the live load applied to the left side of the central buckle is x1, and the length of the live load section on the left side is t1. The starting point for the live load applied to the right side of the central buckle is x2, and the length of the live load section on the right side is t2. Based on the live load position on the stiffening girder, the main span is divided into 6 continuous segments: 0≤x≤x1, x1≤x≤x1+t1, x1+t1≤x≤L / 2, L / 2≤x≤L / 2+x, L / 2+x2≤x≤L / 2+x2+t2, and L / 2+x2+t2≤x≤L. Here, x is any point on the stiffening girder along the longitudinal direction of the bridge within the entire span; L is the total span length. Due to the asymmetry of the horizontal forces of the main cables on both sides of the central buckle, the stiffening girder has a vertical equilibrium equation for each of the 6 segments of the main span. The 5 vertical equilibrium equations are as follows:

[0192]

[0193] in:

[0194] κ1=E g I g +E c I c

[0195] κ2=(H g +H q1 (K3+K4) / 2-E c I c a(K3+K4) / 2

[0196] κ3=(H g +H q1 (-K3+K4) / 2-E c I c a(-K3+K4) / 2

[0197] κ4=(H g +H q2 (K3+K4) / 2-E c I c a(K3+K4) / 2

[0198] κ5=(H g +H q2 (-K3+K4) / 2-E c I c a(-K3+K4) / 2

[0199] In the formula: κ1-κ5 are 5 intermediate computational quantities.

[0200] (3) The stiffening beam deflection w includes 6 components: w1, w2, w3, w4, w5, and w6. Their specific analytical expressions are as follows:

[0201]

[0202] in:

[0203]

[0204]

[0205]

[0206]

[0207] Where: C1-C 24 There are 24 unknown coefficients; η1-η 12 There are 12 intermediate computational quantities; E g and Ig These are the elastic modulus and moment of inertia of the stiffened beam, respectively; q is the uniformly distributed live load value.

[0208] As can be seen from the above, the analytical expression for the deflection of each stiffened beam is H. q1 and H q2 The function has 4 unknown coefficients; there are 6 analytical expressions for stiffening beam deflection, with a total of 24 unknown coefficients.

[0209] The analytical expression for beam end rotation angle is obtained by differentiating the analytical expression for deflection.

[0210] (4) The analytical expression for the beam end rotation angle is:

[0211] The expression for the beam end rotation angle is obtained by differentiating the stiffening beam deflection at the beam end:

[0212]

[0213]

[0214] Step 6: Establish the function of unknown coefficients

[0215] Based on the characteristics that the vertical displacement at the stiffened beam end is 0, the bending moment at the stiffened beam end is 0, the displacement and rotation of the stiffened beam remain continuous, and the shear force and bending moment of the stiffened beam remain continuous, 24 boundary conditions are obtained. Then, the 24 unknown coefficient functions of the 6 deflection analytical expressions in step 5 are solved; each unknown coefficient function is H. q1 and / or H q2 The function.

[0216] The 24 boundary conditions mentioned above are as follows:

[0217] A. When the vertical displacement at the end of the stiffened beam is 0, the following two boundary conditions apply:

[0218] w1(0)=w6(L)=0

[0219] B. When the bending moment at the end of the stiffened beam is 0, the following two boundary conditions apply:

[0220] E g I g w″1(0)=E g I g w″6(L)=0

[0221] C. When the stiffened beam's displacement and rotation remain continuous, the following ten boundary conditions apply:

[0222] w1(x1)=w2(x1)

[0223] w′1(x1)=w′2(x1)

[0224] w2(x1+t1)=w3(x1+t1)

[0225] w′2(x1+t1)=w′3(x1+t1)

[0226] w3(L / 2)=w4(L / 2)

[0227] w′3(L / 2)=w′4(L / 2)

[0228] w4(L / 2+x2)=w5(L / 2+x2)

[0229] w′4(L / 2+x2)=w′5(L / 2+x2)

[0230] w5(L / 2+x2+t2)=w6(L / 2+x2+t2)

[0231] w′5(L / 2+x2+t2)=w′6(L / 2+x2+t2)

[0232] D. When the shear force and bending moment of the stiffened beam remain continuous, the following ten boundary conditions apply:

[0233] w″1(x1)=w″2(x1)

[0234] w″′1(x1)=w″′2(x1)

[0235] w″2(x1+t1)=w″3(x1+t1)

[0236] w″′2(x1+t1)=w″′3(x1+t1)

[0237] w″3(L / 2)=w″4(L / 2)

[0238] w″′3(L / 2)=w″′4(L / 2)

[0239] w″4(L / 2+x2)=w″5(L / 2+x2)

[0240] w″′4(L / 2+x2)=w″′5(L / 2+x2)

[0241] w″5(L / 2+x2+t2)=w″6(L / 2+x2+t2)

[0242] w″′5(L / 2+x2+t2)=w″′6(L / 2+x2+t2).

[0243] Step 7: Solve for H q1 and H q2 Specifically, it includes the following steps:

[0244] Step 7-1, Given H q1 and H q2 The initial value of H in this embodiment is... q1 and H q2 The method for giving initial values ​​is as follows:

[0245] H q1 =H q2 =qL 2 (t1+t2) / h

[0246] Where h is the sag of the stride.

[0247] Step 7-2, Solve for the unknown coefficients: The H coefficients from step 7-1... q1 and H q2 The values ​​are substituted into the unknown coefficient function in step 6 to obtain 24 unknown coefficients.

[0248] Step 7-3, Solve for the stiffened beam deflection w: The H value from step 7-1... q1 and H q2 The value, along with the 24 unknown coefficients obtained from the steps, are substituted into the analytical expression for the stiffening beam deflection in step 5 to obtain the stiffening beam deflection w.

[0249] Step 7-4: Update the main cable compatibility equation: Change K in step 3... g Substituting the expression and the stiffening girder deflection w obtained in step 7-3 into the main cable compatibility equation in step 4, we obtain the updated main cable compatibility equation; at this point, the updated main cable compatibility equation contains only the quantity to be solved, H. q1 H q2 The equation.

[0250] Step 7-5, Solve for H q1 and H q2 The manta ray foraging optimization algorithm is used to solve the updated main cable compatibility equation. When the updated main cable compatibility equation converges, H is obtained. q1 and H q2 The optimal value.

[0251] The manta ray foraging optimization algorithm is a mature existing technology, and the specific optimization steps are as follows:

[0252] (1) Initialize the manta ray particle swarm.

[0253] (2) In the first stage, one of the strategies of chain foraging and whirlwind foraging is randomly adopted for updating.

[0254] (3) In the second stage, the current position is updated by using the somersault strategy.

[0255] (4) Determine whether convergence has occurred. If it has, end the optimization; otherwise, update the particle swarm and return to the first stage.

[0256] Step 8: Solve for deflection and beam end rotation: Based on H obtained in Step 7 q1 and H q2 Find the optimal value, repeat steps 7-2 and 7-3 to obtain the stiffening beam deflection, and differentiate the stiffening beam deflection to obtain the beam end rotation angle.

[0257] The preferred embodiments of the present invention have been described in detail above. However, the present invention is not limited to the specific details in the above embodiments. Within the scope of the technical concept of the present invention, various equivalent transformations can be made to the technical solutions of the present invention, and these equivalent transformations all fall within the protection scope of the present invention.

Claims

1. A method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle under live load, characterized in that: Includes the following steps: Step 1: Adding a central buckle: The suspension bridge has at least one main cable, and each main cable is connected to the stiffening girder through several parallel suspenders; an inverted V-shaped central buckle is set at the mid-span of each main cable; the apex of the central buckle is anchored to the mid-span of the corresponding main cable, and the two bottom ends of the central buckle are anchored to the stiffening girder directly below; the stiffening girders on both sides of each central buckle are tensioned and constrained by horizontal cables. Step 2: Determine the basic alignment of the main cable under dead load considering bending stiffness: Considering the bending stiffness of the main cable, and based on the main cable equilibrium equation under dead load, obtain the basic alignment y of the main cable under dead load. Step 3: Calculate the equivalent spring stiffness of the side cable and the center buckle: The equivalent spring stiffness of the side cable is obtained based on the conservation of stress-free length; the equivalent spring stiffness K of the center buckle is... g It is calculated based on the longitudinal displacement of the midpoint of the main cable, and K g For H q1 H q2 The function of the stiffening beam deflection w; where H q1 and H q2 These are the horizontal increments of the main cable on both sides of the central buckle caused by live load, which are the quantities to be solved. Step 4: Establish the main cable compatibility equations: Based on the deformation relationship of the main cables on both sides of the central buckle in the horizontal projection direction, establish the compatibility equations at both ends of the main cable; where both compatibility equations are H... q1 H q2 and K g The functional equation; Step 5: Establish analytical expressions for deflection and beam end rotation: The starting point of the live load applied to the left side of the central buckle is x1, and the length of the live load section on the left side is t1; the starting point of the live load applied to the right side of the central buckle is x2, and the length of the live load section on the right side is t2; according to the live load position on the stiffening girder, the main span is divided into 6 continuous segments, namely: 0≤x≤x1, x1≤x≤x1+t1, x1+t1≤x≤L / 2, L / 2≤x≤L / 2+x, L / 2+x2≤x≤L / 2+x2+t2 and L / 2+x2+t2≤x≤L; where x is any point on the stiffening girder along the longitudinal direction of the bridge within the entire span; L is the total span length; due to the asymmetry of the horizontal forces of the main cables on both sides of the central buckle, the stiffening girder has a vertical equilibrium equation for each of the 6 segments of the main span; based on the 6 vertical equilibrium equations, 6 analytical expressions for the stiffening girder deflection are obtained; each analytical expression for the stiffening girder deflection is H q1 and H q2 The function has 4 unknown coefficients; the analytical expression for the beam end rotation angle is obtained by differentiating the analytical expression for deflection; Step 6: Establish the unknown coefficient functions: Based on the characteristics that the vertical displacement at the stiffening beam end is 0, the bending moment at the stiffening beam end is 0, the displacement and rotation of the stiffening beam remain continuous, and the shear force and bending moment of the stiffening beam remain continuous, 24 boundary conditions are obtained. Then, the 24 unknown coefficient functions of the 6 deflection analytical expressions in Step 5 are solved; each unknown coefficient function is H. q1 and / or H q2 The function; Step 7: Solve for H q1 and H q2 Specifically, it includes the following steps: Step 7-1, Given H q1 and H q2 The initial value; Step 7-2, Solve for the unknown coefficients: The H coefficients from step 7-1... q1 and H q2 The values ​​are substituted into the unknown coefficient function in step 6 to obtain 24 unknown coefficients; Step 7-3, Solve for the stiffened beam deflection w: The H value from step 7-1... q1 and H q2 The value, along with the 24 unknown coefficients obtained from the steps, are substituted into the analytical expression for the stiffening beam deflection in step 5, and then the stiffening beam deflection w is obtained. Step 7-4: Update the main cable compatibility equation: Change K in step 3... g Substituting the expression and the stiffening girder deflection w obtained in step 7-3 into the main cable compatibility equation in step 4, we obtain the updated main cable compatibility equation; at this point, the updated main cable compatibility equation contains only the quantity to be solved, H. q1 H q2 The equation; Step 7-5, Solve for H q1 and H q2 The manta ray foraging optimization algorithm is used to solve the updated main cable compatibility equation. When the updated main cable compatibility equation converges, H is obtained. q1 and H q2 The optimal value; Step 8: Solve for deflection and beam end rotation: Based on H obtained in Step 7 q1 and H q2 Find the optimal value, repeat steps 7-2 and 7-3 to obtain the stiffening beam deflection, and differentiate the stiffening beam deflection to obtain the beam end rotation angle.

2. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 1, characterized in that: The apex of the central buckle and the corresponding trough of the main cable form anchorage point A. The two bottom endpoints of the central buckle and the stiffening beam directly below form left anchorage point B and right anchorage point C. The two horizontal cables are the left horizontal cable and the right horizontal cable. The left end of the left horizontal cable is anchored to the left bridge tower crossbeam, forming anchorage point D. The right end of the left horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point B. The left end of the right horizontal cable is anchored to the bottom of the stiffening beam corresponding to anchorage point C. The right end of the right horizontal cable is anchored to the right bridge tower crossbeam, forming anchorage point E.

3. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 1, characterized in that: In step 2, the formula for calculating the basic alignment y of the main cable under constant load is: in: In the formula, g g and g c These are the uniformly distributed self-weight loads of the stiffening girder and the main cable, both of which are known quantities; H g The horizontal tension of the main cable under constant load is an unknown quantity, which is determined by boundary conditions. K1, K2, K3, and K4 are all coefficients to be determined, which are obtained through boundary conditions; E c and I c These are the elastic modulus and moment of inertia of the main cable, respectively. 'a' represents intermediate computational quantities.

4. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 3, characterized in that: In step 2, solve for K1, K2, K3, K4 and H. g The five boundary conditions are as follows: In the formula, h is the mid-span elevation; x = 0 represents the hinge point between the left side of the main cable and the left bridge tower; x = L / 2 represents the mid-span point, and L / 2 is the half-span length; x = L represents the hinge point between the right side of the main cable and the right bridge tower.

5. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 3, characterized in that: In step 3, the side cables include a left cable located to the left of the main cable and a right cable located to the right of the main cable; the bending stiffness of the left cable is K. cl The bending stiffness of the right cable is K. cr The equivalent spring stiffness of the central buckle is K. g The corresponding formulas are as follows: in: Δ=(w(L / 2-l h / 2)-w(L / 2+l h / 2)) / 2 In the formula, α is the inclination angle of the left or right cable; L sl It is the horizontal projection length of the left cable; L sr It is the horizontal projection length of the cable on the right. H q1 and H q2 These are the horizontal increments of the main cable on both sides of the central buckle caused by live load, which are the quantities to be solved. A c It is the cross-sectional area of ​​the main cable; l v and l h It is the projected length of the center in both the vertical and horizontal directions; E cb and A cb These are the elastic modulus and cross-sectional area of ​​the central buckle, respectively. E hc and A hc These are the elastic modulus and cross-sectional area of ​​the horizontal cable, respectively. δ c1 It is the longitudinal displacement of the central buckle under the action of the horizontal force of the main cable; δ c2 The longitudinal displacement of the main cable mid-span is caused by the asymmetrical vertical displacement of the stiffening girder on both sides of the central buckle. Δ represents the vertical displacement on both sides of the central buckle; w is the deflection of the stiffened beam.

6. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 5, characterized in that: In step 4, the compatibility equations for the main cables at both ends of the main span are: In the formula, L c1 and L c2 These are all intermediate calculations; w is the deflection of the stiffened beam, a quantity to be solved; K tl and K tr These are the equivalent spring stiffnesses of the left and right bridge towers, respectively, and are known quantities.

7. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 4, characterized in that: In step 5, the stiffening beam deflection w includes six components: w1, w2, w3, w4, w5, and w6. Their specific analytical expressions are as follows: in: κ1=E g I g +E c I c κ2=(H g +H q1 )(K3+K4) / 2-E c I c a(K3+K4) / 2 κ3=(H g +H q1 )(-K3+K4) / 2-E c I c a(-K3+K4) / 2 κ4=(H g +H q2 )(K3+K4) / 2-E c I c a(K3+K4) / 2 κ5=(H g +H q2 )(-K3+K4) / 2-E c I c a(-K3+K4) / 2 In the formula, C1-C 24 There are 24 unknown coefficients; η1-η 12 There are 12 intermediate computational resources; κ1-κ5 are 5 intermediate computational resources; E g and I g These are the elastic modulus and moment of inertia of the stiffened beam, respectively; q is the uniformly distributed live load value.

8. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 7, characterized in that: In step 6, the 24 boundary conditions are as follows: A. When the vertical displacement at the end of the stiffened beam is 0, the following two boundary conditions apply: w1(0)=w6(L)=0 B. When the bending moment at the end of the stiffened beam is 0, the following two boundary conditions apply: HAVE BEEN g I g w″1(0)=E g I g w″6(L)=0 C. When the stiffened beam's displacement and rotation remain continuous, the following ten boundary conditions apply: w1(x1)=w2(x1) w′1(x1)=w′2(x1) w2(x1+t1)=w3(x1+t1) w′2(x1+t1)=w′3(x1+t1) w3(L / 2)=w4(L / 2) w′3(L / 2)=w′4(L / 2) w4(L / 2+x2)=w5(L / 2+x2) w′4(L / 2+x2)=w′5(L / 2+x2) w5(L / 2+x2+t2)=w6(L / 2+x2+t2) w′5(L / 2+x2+t2)=w′6(L / 2+x2+t2) D. When the shear force and bending moment of the stiffened beam remain continuous, the following ten boundary conditions apply: w″1(x1)=w″2(x1) w″′1(x1)=w″′2(x1) w″2(x1+t1)=w″3(x1+t1) w″′2(x1+t1)=w″′3(x1+t1) w″3(L / 2)=w″4(L / 2) w″′3(L / 2)=w″′4(L / 2) w″4(L / 2+x2)=w″5(L / 2+x2) w″′4(L / 2+x2)=w″′5(L / 2+x2) w″5(L / 2+x2+t2)=w″6(L / 2+x2+t2) w″′5(L / 2+x2+t2)=w″′6(L / 2+x2+t2)。 9. The method for determining the vertical deformation of a suspension bridge with horizontal cables and a central buckle according to claim 7, characterized in that: In step 7-1, H q1 and H q2 The method for giving the initial value is: H q1 =H q2 =qL 2 (t1+t2) / h; where h is the sag height at the mid-span.

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